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Let $X$ be a topological space and $\{V_i\}$ a cover of $X$. Let $F_i:\mathsf{Open}(V_i)^{\mathrm{op}}\to \mathsf{Sets}$ be a family of sheaves. One can glue this family to obtain a sheaf $F:\mathsf{Open}(X)^{\mathrm{op}}\to \mathsf{Sets}$, iff the family satisfies some local compatibility conditions like the cocycle condition on threefold unions. This is the situation for sheaves on a topological space.

Is there a similar statement for sheaves on a Grothendieck site $C$, in particular for the site $\mathsf{Sch}$ of schemes with the Zariski- or the etale topology?

Perhaps there is a sheaf $F:C^{\mathrm{op}}\to \mathsf{Sets}$ iff there is a compatible family $F:C^{\mathrm{op}}/V_i\to \mathsf{Sets}$ of sheaves on the slice topoi? What are the compatibility conditions then? Does somebody have a reference?

My second question is most probably nonsense but it would be nice if somebody could support this such that I can forget about this idea forever, even for the situation of sheaves on a topological space: Let $F_i:\mathsf{Open}(V_i)^{\mathrm{op}}\to \mathsf{Sets}$ and $G_i:\mathsf{Open}(V_i)^{\mathrm{op}}\to \mathsf{Sets}$ be two families of sheaves and $F_i\to G_i$ a morphism of sheaves. Does this glue to a morphism $F\to G$ of the glued sheaves?

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I hope you don't mind, I have edited your question a bit so that the categories are given names that are formatted differently than normal math-mode symbols (although I'm sure no one would confuse $Open$ with the product of four symbols named "$O$", "$p$", "$e$", and "$n$", a visual distinction is always nice). Please feel free to undo my edit, or change it some more, if you don't like the way I changed it. –  Zev Chonoles Apr 8 '13 at 14:53
    
Dear @ZevChonoles, thank you for the editing. –  Ronald Bernard Apr 8 '13 at 17:11

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up vote 2 down vote accepted

Yes, a reference is Stacks Project 7.23. The proofs there are quite sketchy. The reason is that they work exactly as for topological spaces. Just replace open subsets by objects of the site, open coverings by coverings in the abstract sense, intersections by fiber products, sections by morphisms, subspace topology by slice topology.

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Ok, so even the second question with the morphisms has a positive answer. I am surprised for the following reason: If you want to construct a morphism between two schemes, you can't do it locally. You can only check locally if a given morphism is an isomorphism. Even in topology, There are for example two non-isomorphic $2$-sheeted covering spaces! What is different in this situation? Sorry but I am confused now. –  Ronald Bernard Apr 8 '13 at 18:14
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Morphisms "always" glue (for topological spaces, ringed spaces, locally ringed spaces, in particular schemes (even in the étale topology) and manifolds). See also the notion of subcanonical Grothendieck topology. –  Martin Brandenburg Apr 8 '13 at 18:33
    
Ok, I have found the mistake in my argument. Thank you. –  Ronald Bernard Apr 8 '13 at 21:22

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